Answers

My parents brought me up with a lot of manners, but that's one that I never heard. I'd so go ahead and keep your hands warm if you want, it's not like the other person is going to expect you to to keep your hands out for propriety's sake.

This is not a rule; it is basic good manners and politeness. If you are close enough to someone to speak to them, then you are close enough to strike them with a dagger. If your hands are out of your pockets, i.e. not concealed, then the other person can see that you are not holding a dagger, therefore you do not intend to do them any harm. This is probably the origin of the belief that you should not conceal your hands when you speak to someone. A similarly-old tradition says that you should not have your hands below the table when you are eating a meal. Keep your hands in view, so that others who are at the same table can see that you are not concealing a knife.

I've never heard of that before. As someone else said, there's situations in which it'd be common courtesy to not keep your hands in your pockets. To some, this could come across as you being casual and maybe even unconcerned, which isn't what you'd want to look like at certain times.